Optimal. Leaf size=256 \[ \frac{1015187 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{8756748 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2}{45} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac{202}{351} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac{13318 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{5/2}}{11583}+\frac{\sqrt{2 x+3} (629153 x+534271) \left (3 x^2+5 x+2\right )^{3/2}}{243243}-\frac{\sqrt{2 x+3} (7817373 x+6006884) \sqrt{3 x^2+5 x+2}}{21891870}-\frac{207851 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{6254820 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.192898, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {832, 814, 843, 718, 424, 419} \[ -\frac{2}{45} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac{202}{351} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac{13318 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{5/2}}{11583}+\frac{\sqrt{2 x+3} (629153 x+534271) \left (3 x^2+5 x+2\right )^{3/2}}{243243}-\frac{\sqrt{2 x+3} (7817373 x+6006884) \sqrt{3 x^2+5 x+2}}{21891870}+\frac{1015187 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{8756748 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{207851 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{6254820 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 832
Rule 814
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx &=-\frac{2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac{2}{45} \int (3+2 x)^{3/2} \left (385+\frac{505 x}{2}\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac{202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac{2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac{4 \int \sqrt{3+2 x} \left (\frac{46155}{4}+\frac{33295 x}{4}\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx}{1755}\\ &=\frac{13318 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac{202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac{2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac{8 \int \frac{\left (242380+\frac{1348185 x}{8}\right ) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt{3+2 x}} \, dx}{57915}\\ &=\frac{\sqrt{3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac{13318 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac{202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac{2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac{4 \int \frac{\left (\frac{17298645}{8}+\frac{13028955 x}{8}\right ) \sqrt{2+5 x+3 x^2}}{\sqrt{3+2 x}} \, dx}{3648645}\\ &=-\frac{\sqrt{3+2 x} (6006884+7817373 x) \sqrt{2+5 x+3 x^2}}{21891870}+\frac{\sqrt{3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac{13318 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac{202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac{2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac{2 \int \frac{\frac{1333245}{2}-\frac{21824355 x}{8}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{164189025}\\ &=-\frac{\sqrt{3+2 x} (6006884+7817373 x) \sqrt{2+5 x+3 x^2}}{21891870}+\frac{\sqrt{3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac{13318 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac{202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac{2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac{207851 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{12509640}+\frac{1015187 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{17513496}\\ &=-\frac{\sqrt{3+2 x} (6006884+7817373 x) \sqrt{2+5 x+3 x^2}}{21891870}+\frac{\sqrt{3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac{13318 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac{202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac{2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac{\left (207851 \sqrt{-2-5 x-3 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 x^2}{3}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{6254820 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (1015187 \sqrt{-2-5 x-3 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 x^2}{3}}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{8756748 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{\sqrt{3+2 x} (6006884+7817373 x) \sqrt{2+5 x+3 x^2}}{21891870}+\frac{\sqrt{3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac{13318 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac{202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac{2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac{207851 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{6254820 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{1015187 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{8756748 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.377207, size = 218, normalized size = 0.85 \[ -\frac{1590604 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+2 \left (630485856 x^9+1907623872 x^8-11776907520 x^7-82311172272 x^6-217661096106 x^5-319887585072 x^4-283276026729 x^3-150475882830 x^2-44206631441 x-5523159638\right ) \sqrt{2 x+3}+1454957 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{131351220 (2 x+3) \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 166, normalized size = 0.7 \begin{align*}{\frac{1}{7881073200\,{x}^{3}+24956731800\,{x}^{2}+24956731800\,x+7881073200}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -12609717120\,{x}^{9}-38152477440\,{x}^{8}+235538150400\,{x}^{7}+1646223445440\,{x}^{6}+4353221922120\,{x}^{5}+3620978\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) +1454957\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) +6397751701440\,{x}^{4}+5665520534580\,{x}^{3}+3009604954020\,{x}^{2}+884278124520\,x+110521391040 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (2 \, x + 3\right )}^{\frac{5}{2}}{\left (x - 5\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (12 \, x^{5} - 4 \, x^{4} - 185 \, x^{3} - 406 \, x^{2} - 327 \, x - 90\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (2 \, x + 3\right )}^{\frac{5}{2}}{\left (x - 5\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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